Synthesized soliton crystals

Dissipative Kerr soliton (DKS) featuring broadband coherent frequency comb with compact size and low power consumption, provides an unparalleled tool for nonlinear physics investigation and precise measurement applications. However, the complex nonlinear dynamics generally leads to stochastic soliton formation process and makes it highly challenging to manipulate soliton number and temporal distribution in the microcavity. Here, synthesized and reconfigurable soliton crystals (SCs) are demonstrated by constructing a periodic intra-cavity potential field, which allows deterministic SCs synthesis with soliton numbers from 1 to 32 in a monolithic integrated microcavity. The ordered temporal distribution coherently enhanced the soliton crystal comb lines power up to 3 orders of magnitude in comparison to the single-soliton state. The interaction between the traveling potential field and the soliton crystals creates periodic forces on soliton and results in forced soliton oscillation. Our work paves the way to effectively manipulate cavity solitons. The demonstrated synthesized SCs offer reconfigurable temporal and spectral profiles, which provide compelling advantages for practical applications such as photonic radar, satellite communication and radio-frequency filter.

basic pulse dynamics, synthesized potential field and oscillated dynamics of soliton crystals.
In Section 3, we provide more details of the experiment, including the four representative SC intracavity power traces all showing direct transition from modulated Turing pattern (TP) and the control-laser based XPM comb generation.

Device characterization
The four-port integrated microcavity (radius = 592.1 µm, FSR = ∼ 48.9 GHz) is based on CMOS-compatible low loss (∼ 0.06 dB/cm at C band), high-index (n=1.6) doped silica platform (1) . The waveguides are surrounded by SiO 2 cladding. During the fabrication process, the core film is deposited using chemical vapor deposition, then the device patterns are printed in photoresist using in-line stepper and etched by reactive ion etching (2) . To improve the light on-chip coupling efficiency, a mode transformer (MT) structure is added to the four ports of the device, which allows optimized coupling loss of ∼ 2.5 dB per facet (3). The loaded Q of our device is ∼ 2.63 million, as shown in Supplementary Figure 1   For simplicity, we only consider the part of the control light which co-propagrates with the pump light. The pump light field can be expressed as E p = E p e −iωpt e iµpφ l , while the control light field can be expressed as E c = E c e −iωct e iµcφ l . E p (E c ) is the complex amplitude of pump (control) light, ω p (ω c ) is the angular frequency of pump (control) light, µ p (µ c ) is the mode number of pump (control) mode, φ l is the polar angle in laboratory coordinate.

Dichromatic-pump field
The total electrical field injected into the microcavity is the sum of the pump field and the control field, which can be expressed as: Here, φ = φ l − D 1 t is the rotating angular coordinate with angular velocity D 1 = 2π × FSR (4). η = µ c − µ p denotes the mode number difference between control light and pump light, and ∆ω = ω c − ω p − ηD 1 is the beat angular frequency or the frequency mismatch. By taking the frequency and mode number of the pump laser as the reference (which eliminates the e −iωpt e iµpφ l factor in Eq. 1), the total injected electrical field is reduced to which oscillates periodically with a frequency of ∆ω/2π. The term E c e −i∆ωt e iηφ exhibits the behaviour of a traveling wave, with frequency ∆ω/2π, spatial periodicity η and traveling speed ∆ω/η.

Dichromatic-pump LLE model
When introducing the additional control light, the LLE can be rewritten as: where an additional pump term is added according to Eq. 2. Here A(φ, t) is the slowly varying intracavity field amplitude defined on the rotating angular coordinate φ = φ l − D 1 t. D 2 is the second-order group velocity dispersion (GVD) term. The input field f p = P p /hω 0 , f c = r eff RP c /hω 0 , where R = 3.27% is the maximum power reflection rate, and r eff denotes the power coefficient related to the control laser detuning. In the simulation, we link this value to the measured transmission spectrum shown in Supplementary Figure 1 Total cavity losses are described with the photon decay rate κ. The Kerr nonlinear coefficient g is defined as g =hω 2 0 cn 2 /n 2 0 V eff , where n 2 is the nonlinear refractive index, n 0 is the effective group refractive index and V eff is the effective optical mode volume.
The above LLE can be normalized for convenience by taking t = 2τ /κ and A = κ/2gψ, where τ is the normalized time, ψ is the normalized optical field: The simulations in the main text are performed by numerically integrates the above equation (Eq. S9) with a split-step Fourier method. We use the measured parameters of the real device from our experiment, which includes: FSR = 48.98 GHz, D 2 /2π = 126.3 kHz, κ/2π = 73 MHz.
The Raman effect and higher order dispersion terms are neglected.

Basic pulse dynamics
By introducing the total drive termF =f p +f c e i(ηφ+ϕ 2 ) e −i∆ ωτ , Eq. 4 can be reduced further as: The intracavity field can be Fourier expanded as where ψ µ denotes the optical field of the µ-th comb line and |ψ µ | 2 is the normalized optical energy or photon number of the µ-th mode.
To study the dynamics of a pulse, we can think of it as a particle and study its motion in the rotating angular coordinate. The optical energy is given in Ref. (5), since the constant (i.e., speed of light in vacuum) relation between energy and mass could be normalized, here we also link this equation to the mass of the pulse: Note that the resonance frequency of the µ-th mode (relative to the pump mode) can be approximated with Taylor expansion therefore, the angular group velocity of the µ-th mode is In the rotating angular coordinate, the normalized relative angular group velocity is 2D 2 µ/κ = 2βµ. Therefore, the momentum of the pulse can be defined as the momentum summation over all optical modes where ϕ(φ, τ ) is the argument of ψ. From Eq. 7 and 10, it is clear that in the spatial-temporal domain, we can define the mass density ρ(φ, τ ) and the momentum density J(φ, τ ) as From Eq. 5 and Eq. 11, we could obtain: The first term on the right hand side represents the density flow, while the second and third term correspond to dissipation and pump driving respectively. When deleting the second and third term (loss and pump) we have which is the one-dimensional mass conservation equation in fluid dynamics. This proves that the definition of mass M and momentum P is reasonable and our dynamic theory is self-consistent.
Furthermore, the velocity field can be defined as For a short pulse stably propagating in the microcavity, the velocity (therefore ∂ϕ/∂φ) around the pulse center can be regarded as a constant, meaning that the optical field of the pulse can be approximated as where φ c = φ c (τ ) is the center position of the pulse which may change over time. Since each part of the pulse is considered to have the same speed, the velocity of the pulse should be expressed as where µ cen ≡ µ µ|ψ µ | 2 / µ |ψ µ | 2 is the spectral center mode number. Therefore we have

Synthesized force field and equation of motion for optical solitons
To study the motion of a pulse, we start by taking the time derivative of Eq. 7 and 10 where M 0 = 2π 0 ψ * f p dφ/4π + c.c. and F = (β/π) 2π 0 dφψ * (−i∂/∂φ)F + c.c.. The first term of Eq. 20 shows the dissipation, while the second term denotes the driving force applied by the control laser.
When there is a single soliton circulating the microcavity, the optical field can be approximately described as ] is the CW background and ψ s is the characteristic of optical solitons. The parameters are B s = √ 2ζ, φ τ = β/ζ, and ϕ s = arccos( √ 8ζ/πf p ). Around the pulse center, the sech term ψ s dominates, while away from the pulse center the CW background dominates. The velocity of the homogeneous CW background is strictly zero (locked by the pump laser), meaning that it does not move together with the sech part. Mathematically speaking, it is easy to verify that meaning that the stationary CW background has no contribution to either the total momentum or the total force applied to the intracavity field. In short, ψ s travels independently on top of the stationary CW background. Therefore, when studying the soliton motion, one should ignore ψ b and focus entirely on ψ s (5). For convenience, from now on we omit ψ b and take ψ = ψ s . We Note that whenf c is small, we have |µ cen | 1, |M − M 0 | M 0 , and |P | M 0 .
According to Eq. 20, the driving force applied to the soliton by the control laser can be written as where F 0 ≡ 2βηB s φ τfc sech(πηφ τ /2) = (2β) 3/2 ηf c sech(πη β/ζ/2). Therefore the potential field can be written as which behaves as a traveling wave with velocity ∆ ω/η in the rotating angular coordinate. Since P = M v = M dφ c /dτ , together with Eq. 19 and 20, we obtain the following dynamic equations Since F is small, |dM/dτ | M , Eq. 28 can be approximated as (to the first order of F ) which is the Eq. 2 in the main text.

Motion of soliton crystal
2.5.1 When the frequency mismatch ∆ ω is small: trapped by the potential field When ∆ ω is small, soliton pulses move together with the potential field (4), meaning that Together with Eq. 29 we obtain Therefore together with Eq. 25, the equilibrium position of the soliton is where k is an integer. As for the ± sign which corresponds to two sets of equilibrium positions, only the one set with (dF/dφ c ) φc=φce < 0 is stable. This set of stable equilibrium positions exists only when where ∆ ω max ∼ 0.01 is the critical value. It is obvious that the stable equilibrium positions are equally spaced by 2π/η, which stabilizes the soliton crystal generated during the resonance scanning process.

When the frequency mismatch ∆ ω is large: forced oscillation
When ∆ ω > ∆ ω max , solitons can no longer move synchronously with the potential field, which is the case in the experiment. Mathematically speaking, when ∆ ω ∆ ω max , to the zeroth order of F 0 , the soliton center position φ c ≈ φ c0 and mass M ≈ M 0 remain constant; To the first order of F 0 , where φ c1 , M 1 ∝ F 1 0 ; To the second order of F 0 , where M 21 , M 22 ∝ F 2 0 ; And so on to higher orders. (Here for the amplitude of each frequency oscillation component, only the lowest order of F 0 is kept.) Now we are going to illustrate these mathematical results by the step iterative method.
By replacing φ c with its zeroth order approximation φ c0 in the expression of F (Eq. 25), Eq.
To the second order of F 0 , Eq. 41 and 42 are reduced to Therefore we have where δ 2 = arctan ∆ ω. For the soliton crystal state consisting of η equally spaced pulses, the phase of oscillation is the same for all soliton pulses, meaning that they move synchronously with each other. Therefore, the soliton crystal state is maintained. The above calculations also indicate that the oscillation frequencies are the integer multiples of ∆ ω, which is in good agreement with both the experiment (Fig. 5d main text) and the simulation (Fig 5e main text) results.

High Repeatability of SC generation
Benefitting from the periodically modulated background field, SC states could be determinis-

Cross-phase modulation (XPM) comb
Once SC is synthesized, the index of the microcavity will be periodically modulated which acts on the control light to form an XPM comb. Thus, a secondary comb could be auto-formed which has the same frequency spacing with the synthesized SC. In a recent work, XPM comb has been demonstrated using orthogonally polarized dual-pump scheme, where the weaker TE seed is modulated by the stronger TM soliton comb, thus the XPM comb could be observed below the threshold of the TE comb generation (6). For our configuration, the pump and control lasers belong to the same polarization mode family, the modulation and XPM comb are expected to be more obvious due to the larger modulation index (7). However, since the XPM comb and SC reside in the same resonances, it is difficult to directly distinguish them using an ordinary OSA. Here, we select a control laser which is about 50-FSR away from the pump laser, the XPM comb coverage spreads out of the soliton spectral range (e.g. the sech 2 fitting envelope) and can be directly observed using an OSA. Supplementary Figure 3 shows the corresponding experimental and numerical spectra, where the comb components beyond the fitting curve are XPM comb spectra. The green trace in Supplementary Figure 3(b) shows the simulated XPM comb, which agrees well with the experimental observation shown in Supplementary Figure   3(a) and helps us better resolve the specific comb lines. Note that for simplicity, the group velocity mismatch (GVM) is omitted here. In fact, the GVM only affects the relative power of the two wings (indicated by the black arrows in Supplemetary Figure 3(b)) of the XPM comb spectrum (7). The temporal traces of soliton comb and XPM comb are shown in Supplementary   Figure 3(c), where the soliton and XPM pulse travel together with a relative velocity compared with the lattice traps. It should be noted that recent studies have shown that the XPM comb could be used for broadening the spectral range (8,9).